Special Colloquium

Sarah Koch
Harvard University
An algebraic fingerprint for postcritically finite rational maps
Abstract: In the 1980s William Thurston established his topological characterization of rational maps, one of the central results in the field of holomorphic dynamics. This theorem applies to postcritically finite rational maps (a rational map is postcritically finite if the orbit of every critical point is finite). Given such a rational map, one can define a holomorphic endomorphism of a Teichmueller space associated to it; this endomorphism is called the Thurston pullback map. With the exception of one class of examples, this endomorphism has a unique fixed point, and the eigenvalues of the derivative at this fixed point are all *algebraic*. What do these eigenvalues mean? Do they have any geometric significance in the moduli space of rational maps? In the dynamical plane of the map itself? What algebraic numbers arise this way? We establish some facts about these eigenvalues, and we prove there are no "small eigenvalues" in the case of quadratic polynomials. The general situation is still quite mysterious.
Tea at 4
Wednesday November 28, 2012 at 3:00 PM in SEO 636
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